Cloaking a qubit in a cavity

Cavity quantum electrodynamics (QED) uses a cavity to engineer the mode structure of the vacuum electromagnetic field such as to enhance the interaction between light and matter. Exploiting these ideas in solid-state systems has lead to circuit QED which has emerged as a valuable tool to explore the rich physics of quantum optics and as a platform for quantum computation. Here we introduce a simple approach to further engineer the light-matter interaction in a driven cavity by controllably decoupling a qubit from the cavity’s photon population, effectively cloaking the qubit from the cavity. This is realized by driving the qubit with an external tone tailored to destructively interfere with the cavity field, leaving the qubit to interact with a cavity which appears to be in the vacuum state. Our experiment demonstrates how qubit cloaking can be exploited to cancel the ac-Stark shift and measurement-induced dephasing, and to accelerate qubit readout. In addition to qubit readout, applications of this method include qubit logical operations and the preparation of non-classical cavity states in circuit QED and other cavity-based setups.


INTRODUCTION
Cavity and circuit QED explore light-matter interaction at its most fundamental level, providing the tools to control the dynamical evolution of single atoms and photons in a deterministic fashion [1].This has allowed circuit QED to emerge as a platform to explore the rich physics of quantum optics in novel parameter regimes and to become a leading architecture for quantum computing [2].In this system, strong drives on the cavity are used to realize multi-qubit gates [3,4], to stabilize quantum states of the cavity [5][6][7][8], and for qubit readout [2,9].However, even under moderate cavity photon populations, cavity drives often lead to undesired effects such as qubit transitions [10] resulting in reduced readout fidelity [11], increased dephasing [4,7], and imperfect quantum state stabilization [5,6,8].Other consequences of cavity drives in the dispersive qubit-cavity regime are the qubit ac-Stark shift, which can result in unwanted phase accumulations [12], measurement-induced dephasing [13,14], and Kerr nonlinearity [15].
In this work, we introduce a simple approach to engineer light-matter interaction in a driven cavity and prevent some of these unwanted effects.We show how an appropriately tailored drive on the qubit can decouple the qubit state and the cavity's photon population from one another, resulting in both systems interacting only through vacuum fluctuations of the cavity field.This qubit cloaking mechanism can be exploited to prepare non-classical states of the cavity field.Moreover, in the the qubit in the presence of a cavity photon population, something which we exploit to accelerate qubit readout.Here, we experimentally demonstrate qubit cloaking using a transmon qubit [16] coupled to a coplanar waveguide resonator (Fig. 1.c).

RESULTS
When loaded with a coherent state, the cavity field acts as an effective classical drive on the qubit.A simple intuition behind qubit cloaking is that an additional drive on the qubit can be designed to interfere destructively with this effective drive, resulting in an empty cavity from the perspective of the qubit.As an illustration of this concept, consider a transmon qubit coupled to a microwave cavity, see Fig. 1a,b for a schematic representation.Without driving, the system Hamiltonian reads [2] Ĥ0 = 4E C n2 tr − E J cos φtr + ℏω r â † â + iℏgn tr (â † − â), (1) with ntr and φtr the transmon charge and phase operators, E C the charging energy, E J the Josephson energy, â( †) the annihilation (creation) operator of the cavity mode of frequency ω r , and g the transverse coupling rate.The cavity drive is Ĥ1 = iℏE 1 (t)(â † − â), where E 1 (t) = ε 1 (t) sin(ω 1 t + ϕ 1 ) is the amplitude of the cavity drive at frequency ω 1 with envelope ε 1 (t).The cloaking is triggered by a cancelling drive E 2 (t) on the transmon qubit, which leads to an additional term Ĥ2 = ℏE 2 (t)n tr , such that the total system Hamiltonian is Ĥ = Ĥ0 + Ĥ1 + Ĥ2 .The appropriate choice of E 2 (t) that cloaks the qubit from the cavity field is revealed by moving to a displaced frame using the transformation D(α t ) = exp α t â † − α * t â under which D † (α t )â D(α t ) = â + α t .Choosing α t = t 0 dτ E 1 (τ )e −iωr(t−τ ) has the effect of cancelling the cavity drive Ĥ1 in the displaced Hamiltonian, resulting in an effectice drive on the qubit of the form ig(α * t −α t )n tr owing to the qubit-cavity coupling.Therefore, taking the amplitude of the cancelling tone to be the opposite, disables any drive term in the displaced frame, where the total Hamiltonian comes down to (see Supplementary Note 1 for more details [17]) In short, despite the presence of the drive E 1 populating the cavity with an average photon number |α t | 2 , the qubit experiences the cavity as if it was in the vacuum state, only coupling to vacuum fluctuations of the cavity field.Note that no approximations have been made to arrive at this result which is valid irrespective of the qubitcavity detuning and for arbitrary time-dependent drive  amplitude ε 1 (t) and frequency ω 1 .Beyond the transmon qubit, this result is valid for two-level systems and any nonlinear system linearly coupled to a cavity.The results are unchanged in the presence of qubit decay or dephasing, and the above derivation can exactly account for the finite decay rate κ of the cavity by making the change ω r → ω r − iκ/2 in the expression for α t .In the rotating-wave approximation and for a constant ε 1 , the amplitude α t entering the expression for E 2 in Eq. ( 2) is (ε 1 e −i(ω1t+ϕ1) /2)/(ω r − ω 1 − iκ/2) in steady state.See Supplementary Note 1 [17] for details.This approach relies on two distinct driving ports-one port dedicated to exciting the qubit or nonlinear mode and a second port for driving the cavity mode-and is directly applicable in several cavity-based platforms including semiconducting quantum dots coupled to microwave resonators [18][19][20][21], electrons on solid neon [22], circuit quantum acoustodynamics [23], and trapped atoms in cavity QED [24].While it is an exact and robust result, potential limitations of qubit cloaking are discussed in the Supplemental Note 3 [17].
As a first example, we consider a situation where the qubit is resonant with the cavity, and model the transmon as a two-level system.In the absence of any drives, initializing the qubit in its first excited state and the cavity in the vacuum state leads to vacuum Rabi oscillations at the frequency 2g ′ [1], where g ′ = (g/2)(E J /2E C ) 1/4 is the Jaynes-Cummings coupling [2].This is illustrated in Fig. 2a which shows the results (full black and red lines) of integration of the system's master equation including cavity decay, see Supplementary Note 1 for details [17].With the cavity drive and the cancelling tone present, the cavity population (orange dashed lines) increases following |α t | 2 (dashed-dotted grey line) on top of which oscillations are observed.On the other hand, the qubit population (light blue dashed line) is identical to that observed in the absence of the drives.In other words, instead of collapse and revival which are expected in the presence of a coherent state in the cavity [25,26], here the qubit undergoes oscillations at the vacuum Rabi frequency.The same conclusion holds when initializing the cavity mode in the Fock state |1⟩ and the qubit in the ground state (not shown).This is a clear illustration that under cloaking the qubit only couples to vacuum fluctuations of the cavity.In related work, Alsing et al. [27] have shown how a drive on the atom can suppress atomic fluorescence in the steady-state of a lossless cavity.In contrast, cloaking is an exact result which is valid at all times and in the presence of loss.Moreover, this approach can be used to prepare non-classical states of the cavity field.For example, displaced Fock states D(α) |n = 1⟩ with arbitrary α can be prepared by starting with the same initial state as in Fig. 2a, and waiting for half-integer Rabi periods under appropriate cavity drive amplitude and phase.The Wigner function of the displaced Fock state is plotted at 3/2 Rabi periods in Fig. 2b, and is compared to that of the coherent state at 1 Rabi period.As a second example, consider the typical situation of a dispersive qubit readout where the frequency of the first drive is resonant with the cavity and the qubitcavity detuning is large compared to g ′ such that the system is in the dispersive regime [1,2].In the absence of the cancellation tone, the cavity drive results in ac-Stark shift and broadening of the qubit [13,14].This is illustrated in the top panel of Fig. 3a which shows the numerically computed magnitude of the qubit's absorption spectrum S(ω).For simplicity, this is obtained in the two-level approximation for the transmon where the absorption spectrum takes the form S(ω) = (1/2π) +∞ −∞ e iωt ⟨σ − (t)σ + (0)⟩ s with the subscript s indicating that the average is taken in steady-state.The different colored lines correspond to different cavity drive amplitudes ε 1 .The vertical gray line indicates the bare qubit frequency.In contrast, the bottom panel of Fig. 3a shows the same quantity obtained with the cancellation tone present.In this case, the results for the different drive amplitudes collapse on each other.Because cloaking does not affect vacuum fluctuations, the Lamb shift (i.e. the offset from the bare qubit frequency) remains unchanged at all drive amplitudes.Moreover, the independence of the qubit linewidth on drive power indicates both the absence of measurement-induced dephasing and that Purcell decay remains at its zero-photon value [28,29] under cloaking.
This prediction is experimentally tested on the device shown in Fig. 1c.It consists of a transmon qubit capacitively coupled to a λ/4 microwave coplanar waveguide cavity which is driven through port 1 (E 1 ) via a Purcell filter acting as a bandpass filter at the cavity frequency.The cancellation drive (E 2 ) is applied to port 2, which is weakly capacitively coupled to the transmon.The readout frequency is ω 1 /2π = 7.6648 GHz and cavity decay rate κ/2π = 10.1 MHz.The transmon qubit is capacitively coupled to the cavity, with χ/2π = −2.54MHz the full dispersive shift [2].The qubit has a dressed frequency ωq /2π = 4.962 16 GHz and coherence times T 1 = 25 µs and T 2 = 7.5 µs.
Using Ramsey interferometry, it is possible to determine the decoherence rate and the qubit frequency, and thus extract the increase in dephasing rate δΓ and the ac-Stark shift δω as a function of drive strength.The blue dots in Fig. 3b are obtained in the absence of cancellation drive and show the expected linear increase with drive power [13,14].The full lines are obtained from numerical integration of the system's master equation including cavity loss and are used to calibrate the attenuation factor between the drive power at room temperature and ϵ 2  1 for all experimental plots.The red dots correspond to the measured ac-Stark shift and increase in dephasing rate in the presence of the cancellation tone.As expected from the above discussion, with the properly tailored cancellation drive, they are both suppressed at all cavity drive powers.The cancellation drive follows the analytical expression Eq. ( 2) for E 2 (t) but the attenuation and electrical delay of the line driving port 2 needs to be taken into account in order to relate the complex drive amplitude at room temperature E room 2 (t) to the driving Arm and release qubit readout.a, Full-cosine numerical simulation of the phase-space trajectories of the cavity field in standard dispersive readout, i.e., square pulse, (dashed line) and the arm and release readout approach (full lines) with an arm-step coherent state α = 2.8i.The path of the arming step is here chosen to reproduce the experiment of panel b, but it can be tailored at will.The coloured dots indicate different times in the evolution.In both readout approaches, the qubit is prepared in the ground state and a π pulse is applied (red lines) or not (blue lines) before the readout step.Simulation parameters are the same as in Fig. 3b.b, Experimental qubit measurement error obtained from the overlap between the distributions of the accumulated heterodyne signal over 10 6 repetitions of the experiment.Full line: arm-and-release approach.Dashed line: standard dispersive measurement.c, Arm-and-release pulse sequence used to obtain panels a and b.The arm phase (t < 0) is absent in the case of dispersive readout.
The prefactor relating the two is found experimentally by varying the phase ϕ and amplitude µ to minimize |δΓ + iδω| (see Supplementary Note 2 [17]).This technique thus provides a tool to calibrate the attenuation from room temperature to the qubit port.
The above results suggest a strategy to speed up the dispersive qubit readout.Several approaches have been explored to improve readout fidelity by speeding it up [30][31][32][33][34][35][36].Here, we propose to use a two-step 'arm and release' approach.With the cavity in the vacuum state, the arming step consists in driving the cavity in the presence of the cancelling tone.Because the qubit is uncoupled from the cavity's classical field, this causes a displacement of the cavity field by α t that is independent of the qubit-state.Thus during the pre-arming, the cavity can be stabilized in any coherent state without affecting the qubit.The release step can start at any time after the desired measurement photon population is reached: The cancellation tone is turned off, at which point the qubit couples to the cavity field resulting in a qubit-state dependent rotation of the latter in phase space under the dispersive qubit-cavity coupling.Homo-dyne or heterodyne detection then completes the qubit readout in the usual way [2].
Crucially, upon release the two coherent states |α g,e ⟩ corresponding to the qubit ground and excited states separate as under longitudinal coupling at short times.Indeed, as illustrated in Fig. 4a, after the arming step these coherent states move away from each other in the phase space of the cavity mode (full lines), thus maximizing the qubit measurement rate κ|α g − α e | 2 /2 [14,36].This is reminiscent of previous experiments [33,34] where longitudinal-like separation is obtained with an initially empty cavity using an approach based on the dispersive approximation.In contrast, we emphasize that qubit cloaking can operate at arbitrary drive strength and qubit-cavity detuning.Figure 4a also shows the evolution in phase space of the amplitude of the readout cavity for the usual dispersive readout (dashed line), where there is initially poor separation between the qubit-statedependent coherent states.Similarly to longitudinal readout [36], for the same steady-state photon population (see yellow dots corresponding to t = 10/κ), the state separation is significantly larger at short times in the arm-and-release approach than in the usual dispersive readout (see the coloured dots corresponding to times t = (1, 2, 3)/κ).
The reduction in measurement time provided by the arm-and-release approach is illustrated in Fig. 4b which shows the measurement error versus integration time for the arm-and-release approach (full line) and the standard dispersive readout (dashed line) obtained using the device of Fig. 1c.In these experiments, the qubit is first prepared in the ground state |g⟩ using measurementbased feedback with the usual dispersive readout.As illustrated in Fig. 4c, to obtain the full line in panel b the cavity is then pre-armed with a drive of amplitude ε 1,arm /2π = 95.5 MHz (full blue line) and a cancellation drive ε 2 (dashed dark blue line) for a time τ arm = 1 µs.At the time labelled t = 0, the signal is then integrated for a time τ m with a cavity drive ε 1,ro /2π = 63.7 MHz.The arming cavity drive and the cancellation tone are omitted to obtain the dashed line in panel b.A π pulse of width 25 ns (orange dashed line) is applied or not at the end of the arming step to prepare the state |e⟩ or |g⟩.Interestingly and as explained below, this can be done with high fidelity despite the cavity photon population.Finally, the measurement error is obtained by computing the overlap between the distribution of the accumulated heterodyne signal over N = 10 6 repetitions of the experiment where the qubit is prepared either in |g⟩ or |e⟩ (see Supplementary Note 2 [17]).For a 196 ns integration time, we obtain an average fidelity F = 1 − [P (g|e) + P (e|g)]/2 = (99.35± 0.14) % where P (g|e) = (1.07 ± 0.14) % is the error probability to measure state g when state e was prepared, while P (e|g) = (0.23 ± 0.14) % is the error probability to measure state e when state g was prepared.The finite number of repetitions N results in the uncertainty ±0.14 %.The 0.65 % average error is mostly explained by wrong X gate error FIG. 5. Gate error.Measured error (dots) on qubit X gate (randomized benchmarking) as a function of the steady-state cavity photon number ns when the qubit is cloaked (red) or not (blue).The control drive on the qubit is optimized in the absence of cavity drive and cancellation tone to maximize the gate fidelity, and kept identical for all further measurements.The corresponding measured average X gate error and its error bar at zero cavity drive amplitude are represented as a horizontal gray line and gray area.The dashed red line is the average gate error under cloaking obtained from numerical simulations including the Purcell filter (see Supplementary Note 1 [17]).Error bars account for statistical uncertainty.
In optimizing readout fidelity, it is important to account that the cavity responds at different frequencies in the absence or presence of the cancellation tone.Without cloaking, the qubit-state-dependent steady-states coherent state amplitude is , where ωr is the dressed cavity frequency and χ i is the dispersive qubit-cavity coupling for qubit state i = {g, e} [2].On the other hand, with cloaking the cavity responds as if there was no qubit with the steadystate value α s = (ε 1 e −iϕ1 /2)/(ω r − ω 1 − iκ/2), where now ω r is the bare cavity frequency.Depending on the application, such as optimizing the longitudinal-like nature of the readout, an optimal coherent state can be prepared by adequately choosing ε 1 and ϕ 1 during the arming phase, and changing these quantities as desired during the release phase; see Fig. 4c for the pulse sequence and phase used here for readout.The arming phase-space path shown in Fig. 4a is chosen to replicate our experiment, but it can be tailored.For example, a straight path is obtained by arming the cavity with a drive frequency ω 1 = ω r .
As an additional benefit, since the cloaked qubit does not undergo ac-Stark shift or measurement-induced dephasing (see Fig. 3), it is possible to apply qubit gates during the cavity arming step despite the presence of measurement photons in the cavity.As a result, the time needed to fill the cavity does not factor in the measurement time.To test this idea, we use randomized benchmarking [37,38] on the device of Fig. 1c, and extract the gate error for a qubit π-pulse (X gate) performed in the presence of a readout cavity drive, and in the absence or presence of the cancellation tone, see Fig. 5.As expected, in the absence of cancellation (blue dots) the gate error increases rapidly with the cavity drive amplitude and saturates at a gate error of 0.5, corresponding to the largest gate error which can be reported by randomized benchmarking [39].In the presence of cancellation (red dots), the gate error remains at its coherence-limited value (full gray line) despite the presence of measurement photons in the readout cavity.The red dashed line corresponds to the result of numerical simulations accounting for the Purcell filter of the gate under cloaking, see Supplementary Note 1 [17].Importantly, the qubit control drive is identical for all data points shown in Fig. 5.For large drive amplitudes, the X gate error under cloaking starts to increase (last three points in Fig. 5).This is explained by imperfect experimental calibration of the cancellation tone which can be affected by low-frequency drifts.

DISCUSSION
Qubit cloaking can readily be implemented in current circuit QED experiments.This approach is valid irrespective of the driving frequency and waveshape, and applies to arbitrary qubit-cavity detuning.In the resonant regime, cloaking can be used to prepare non-classical states of the cavity [40,41].In the dispersive regime, cloaking can be used to speed-up qubit readout [42] something which, for example, can be used to shorten quantum error correction cycles in circuit QED-based devices [43][44][45].At the heart of this acceleration is the fact that logical gates can be applied to the qubit while the cavity is armed for readout.Similar ideas can be exploited to speed up two-qubit gates that are assisted by a cavity drive, such as the resonator-induced phase gate [3] or to reduce measurement cross-talk errors in multiplexed readout [46].Moreover, the possibility to cancel large ac-Stark shifts can be leveraged to avoid unwanted phase accumulations, such as in the preparation of bosonic codes states [12].We expect that qubit cloaking will become a useful element in the toolbox of circuit QED, extending beyond the transmon qubit [47].Furthermore, it is applicable to other cavity QED setups where a quantum system linearly couples to a cavity [18,20,21,23,24,48].
We move to the displaced frame ρD (t) = D † (α t )ρ(t) D(α t ) using the displacement operator D(α t ) = exp α t â † − α * t â .In this frame, the evolution of the system state is governed by the equation where we have introduced the transmon Hamiltonian Ĥtr = 4E C n2 tr − E J cos( φtr ).The form of the timedependent complex amplitude α t is chosen such as to cancel the cavity drive.For this, the last two lines in Supplementary Equation (2) must vanish and we thus enforce or, equivalently, as mentioned in the discussion following Eq.(3) in the main text.In this frame, the cavity drive is then effectively passed to the qubit, see the third line of Supplementary Equation ( 2).The latter is cancelled if we * These authors contributed equally to this work. choose The initial condition α 0 can be set to zero; it has no consequence on cloaking and, in this way, E 2 (t) starts from zero at t = 0. Also worth noting is that accounting for thermal incoherent excitations in the cavity-replacing κD[â]ρ by (n th + 1)κD[â]ρ + nth κD[â † ]ρ in Supplementary Equation (1), where nth is the thermal population-, qubit decay and dephasing, or incorporating the multimode nature of the cavity, does not change the effect or form of the cancellation drive.We come back to incoherent cavity excitation in Sec.K.
For a constant cavity drive amplitude ε 1 turned on at t = 0, the cancellation tone takes the form where ω ± = ω r ± ω 1 , ϕ ± = arctan(−2ω ± /κ), and ϕ 1,± = ϕ 1 + ϕ ± .With ω r ≫ κ and taking ω 1 ∼ ω r , the leading terms are the ones in the second line of Supplementary Equation ( 6) and the cancellation tone can be approximated by with In experiments, a cancellation tone ansatz of the form Supplementary Equation (7) can be used where A − , ϕ 1,− , and ϕ − are unknown parameters that can be optimized in a Ramsey-like experiment by minimizing the ac-Stark shift and measurementinduced dephasing, see Fig. 3 in the main text.Importantly, the use of the cancellation tone in Supplementary Equation ( 6) or (7) demands knowledge of the bare cavity frequency ω r .An incorrect estimate for ω r will lead arXiv:2211.05758v2[quant-ph] 10 Oct 2023 to imperfect cancellation, but only for transient times before the terms oscillating at ω r in Supplementary Equation ( 6) or (7) become negligible due to the exponential decay e −κt/2 .
In practice, there can be small crosstalks by which the drive on the cavity (qubit) also weakly drives the qubit (cavity).Formally, for a known E 1 (t), the simple form of E 2 (t) = −2gIm[α t ] which achieves the cancellation is replaced in this case by a linear Volterra equation of the second kind: E 2 (t) = f (t) + t 0 dτ K(t, τ )E 2 (τ ) with a separable Kernel K(t, τ ) = j g j (t)h j (τ ).This equation is guaranteed to have a unique solution.In practice, the experimental calibration of the cancellation tone's amplitude and phase deals with this potential crosstalk.

B. Two-states and RWA approximations for the transmon
In the two-level approximation for the transmon and using the rotating wave approximation (RWA), the Hamiltonian Ĥ0 + Ĥ1 becomes In terms of the transmon parameters [2], the qubit frequency is given by ℏω q ≈ √ 8E C E J −E C and the coupling is g ′ ≈ (g/2)(E J /2E C ) 1/4 .With these approximations, the cancellation tone on the qubit takes the form with To obtain the results of Fig. 2 and Fig. 3a of the main text, we numerically integrate the master equation of Supplementary Equation ( 1) with the Hamiltonians of Supplementary Equation ( 8) and Supplementary Equation (9).

C. Cloaking of a general multilevel quantum system
As is made clear by the above two examples, as long as the coupling Hamiltonian is linear in the cavity mode creation and annihilation operators (â and â † ), the approach proposed here is applicable to a wide range of quantum systems where two driving ports couple asymmetrically to the cavity and the quantum system.Indeed, we can generalize our cloaking method for any multilevel quantum system with Hamiltonian Ĥq , coupled to a cavity with Ĥg = g Ôq (e −iϕ â † + e iϕ â) via any quadrature (i.e.arbitrary phase ϕ), where Ôq is any operator on the multilevel system.
For this general case, the cancellation term Ĥ2 (t) in the Lindblad equation is given by with α t as in Supplementary Equation (4).

D. Inclusion of the Purcell filter
Considering a general multilevel system with Hamiltonian Ĥq as in the previous subsection, we now account for the presence of a Purcell filter cavity coupled to the readout cavity, such that the total Hamiltonian reads where f ( †) is the annihilation (creation) operator of the Purcell mode of frequency ω f , and now the drive is on the Purcell cavity instead of the readout cavity.
In the presence of photon loss at the Purcell filter cavity, the master equation takes the form Following the above approach, we now move to a displaced frame ρD (t) = D † r (α t ) D † f (β t )ρ(t) Dr (α t ) Df (β t ), where the subscripts r and f indicate that the displacement operator acts on the Hilbert space of the readout cavity mode â or the Purcell filter cavity mode f , respectively.Moreover, we choose α t and β t such that they follow the coupled equations of motion With this choice, the drive on the Purcell cavity is cancelled and effectively appears on the qubit.This effective drive on the qubit in the displaced frame is cancelled by the Hamiltonian Ĥ2 (t) in Supplementary Equation (12), with α t now the solution to Supplementary Equation ( 15).

E. Gates and imperfect cancellation
Here we report on the simulation of logical gates on the transmon in the presence of a cavity drive.While the gate fidelity can be maintained constant for any cavity drive amplitude if the exact cancellation tone is used, here we explore the deviations that could arise if the cancellation tone is inexact.
As an example, consider the approximate cancellation in Supplementary Equation (7) which is very precise for ω 1 ∼ ω r ≫ κ.Among other kinds of errors, here we consider that resulting from using the simpler ansatz and study how the gate fidelity deviates from the ideal result.
We consider the Lindblad equation where Ĥ includes the transmon qubit and cavity Hamiltonian Ĥ0 , the cavity drive Ĥ1 , and the cancellation Ĥ2 (if present).The gate consists in a π-rotation of t g = 25 ns duration.Using DRAG to eliminate leakage [3], the gate Hamiltonian reads where f (t) = π 2 (t − t g /2)/(t g /4), ωq the transmon g − e transition frequency dressed by the cavity, and {ε g , d g , ϕ g } the set of parameters which are optimized to realize high-fidelity operations.This optimization is performed only once in the absence of cavity drive and cancellation tone.We note that this waveform for the π pulse is the same as the one we use in our experiment.
Numerically, to characterize the gate fidelity, we use [4] which averages over the six cardinal states (i = ±x, ±y, ±z) of the Bloch sphere in the dressed computational subspace { g, 0 , e, 0 }, corresponding to the qubit logical states dressed by the cavity [5].The operator Û is the X Pauli operator in this subspace and E tg is the dynamical map which evolves the state according to the master equation of Supplementary Equation (17).Using E J /h = 84.28EC /h = 16.826GHz, ω r /2π = 7.657 GHz, g/2π = 140.6MHz, κ/2π = 10.1 MHz, and accounting for three transmon energy levels, we obtain the optimal values ε g /2π ≈ 28.51 MHz, d g ≈ 0.09, and ϕ g /2π ≈ −0.027, producing a gate error E = 1 − F no drives ≈ 0.003 in the absence of cavity and cancellation tones.This value corresponds to the coherence limit (∼ 1−e −γκtg/2 ) set by the Purcell decay rate γ κ [5].
In the presence of the cavity drive and in the displaced frame introduced in the main text and in Supplementary Equation (2), the gate realizes a high-fidelity X πrotation within the subspace { g, 0 , e, 0 } as long as the qubit is cloaked.Back in the laboratory frame, the dynamical evolution starting with the qubit in an arbitrary state and the cavity in the vacuum state, turning on the cavity drive and the cancellation tone, and performing the logical gate corresponds to the transformation ψ g g, 0 + ψ e e, 0 → D(α tg )(ψ g e, 0 + ψ e g, 0 ).(20) To account for the displacement operator, the expression for the gate fidelity in Supplementary Equation (19) in the presence of drive and cancellation is modified to where Note that for ε 1 = 0 (and thus E 2 = 0), α tg = 0 and this expression corresponds to Supplementary Equation (19).
In Supplementary Figure 1a we show the average gate error obtained from the above expression and integration of the master equation of Supplementary Equation ( 17) as a function of ε 1 when the cancellation tone is off (blue line) and with the exact cancellation tone (red line).In the latter case and as expected from the discussion in the main text, the gate error is constant at ∼ 0.3% for all drive amplitudes but rapidly increases with drive amplitude in the absence of cancellation.In Supplementary Figure 1b the gate error is plotted considering the approximate cancellation tone in Supplementary Equation (7), but in the hypothetical case of an imprecise estimate of the value of ω r (̸ = ω exact r ), highlighting the importance of correctly determining the bare cavity frequency to use in the cancellation tone.We reiterate that the imprecision leads to a slightly incomplete cancellation which is corrected after a short transient due to the exponential decay in time of the driving terms at the bare cavity frequency.When necessary, the bare frequency ω r could be a parameter to optimize over in the experimental calibration of the cancellation tone.In Supplementary Figure 1c and d, we show the error when we use the cancellation tone in Supplementary Equation ( 6) but with phase and amplitude relative offsets, respectively.In this case, we replace ϕ ± → ϕ ± (1 + δϕ) or ε 1 → ε 1 (1 + δε) in the equation for E 2 (t), taking ϕ 1 = 0 for simplicity.In the case of phase offset, the main contribution to the deterioration of the gate comes from the offset in ϕ − , which enters in the leading terms (second line in Supplementary Equation ( 6)).Finally, in Supplementary Figure 1e, we show the result of using the simpler anzat in Supplementary Equation ( 16) with a single frequency ω 1 , showing that when ω 1 ≈ ω r the gate maintains a high fidelity and deteriorates as the cancellation tone frequency shifts away.Supplementary Figure 1.Numerical X-gate average error.a, Error as a function of cavity drive amplitude without cancellation (blue line) and with exact cancellation (red line).b, Error when the cancellation tone in Supplementary Equation ( 7) is used, but with a hypothetical imprecise estimate of the cavity frequency value, ωr ̸ = ω exact r .Different colors correspond to ε1/2π = 1, 5, 10, and 15 MHz.The arrow indicates increasing values of ε1.c,d, Error when there is, respectively, a phase (δϕ) or amplitude (δε) relative offset (see text).Same ε1 values as in b.In all a,b,c,d, the cavity is driven at the readout frequency, ω1 = ωRO = ωr + (χg + χe)/2.e, The simpler cancellation tone ansatz in Supplementary Equation ( 16) is used, which consist of a single frequency ω1.The drive amplitude is ε1/2π = 15 MHz.The vertical dashed line indicates the value of the readout frequency, ωRO.The value of all parameters are indicated in the text.

F. Gate simulation including Purcell cavity
Here we give details about the simulations used to extract the calculated average π-rotation (X gate) error results shown in Fig. 5 in the main text, for the case of a cloaked qubit.Below we also show results for this error when the qubit is not cloaked.
Starting from Supplementary Equations ( 13) and (14) for the case of the transmon, i.e.Ĥq = Ĥtr and Ĥg = ign tr (â † − â), we first diagonalize the Hamiltonians of the readout and Purcell cavities.In this basis, they become hybridized.Since ω r ∼ ω f ≫ |J|, we use the rotatingwave approximation in the cavity-cavity coupling.Moreover, adding intrinsic transmon decay and dephasing, we arrive at with Ĥgate given in Supplementary Equation ( 18), d = j √ j |j⟩ ⟨j + 1| is a lowering operator in the energy eigenbasis of the transmon, Ĥtr |j⟩ = E j |j⟩, and where with tan θ = 2J/(ω r − ω f ) and The full set of simulation parameters that we use are shown in Supplementary Table I.With these values of γ and γ ϕ , we obtain a qubit lifetime and Ramsey coherence time of T 1 = 25 µs and T 2 = 7.5 µs, respectively, as in our experiment.
For the numerical simulations of the X gate, we work in the displaced frame where the drive on the Purcell cavity is effectively passed to the qubit.This allows us to integrate the master equation with truncated Hilbert spaces for the modes â± that are smaller than would be required if we worked in the laboratory frame.In the results shown in Supplementary Figure 2   Error, 1 Supplementary Figure 2. Predicted X-gate average error accounting for the Purcell cavity.As a function of the number of photons in the readout cavity at the end of the gate, the blue curve with dots shows the error without cancellation, and the dashed line the error with cancellation.The minimum error attained is ∼ 0.163%.Parameters are indicated in Supplementary Table I.
in the main text), we use the definition of Supplementary Equation (21), where the six cardinal states of the Bloch sphere are now those of the logical subspace { g, 0 + , 0 − , e, 0 + , 0 − }, corresponding to the ground and excited states of the qubit dressed by the hybridized modes â± (here 0 ± is the zero Fock state of these modes).
The gate error obtained from numerical simulations quantitatively agree with experiments in the presence of cloaking (see dashed red line in Fig. 5 in the main text and Supplementary Figure 2).In the absence of cloaking, full blue line in Supplementary Figure 2, we find numerically the expected increase of the gate error with cavity photon number but the agreement with the experimental observations is not quantitative.While the average gate error in the presence of cloaking is coherence limited and thus simple to fit to the experimental results by adjusting γ and γ ϕ , in the absence of cloaking the error depends on the quality of the fit of the bare parameters used in the model, as well as on a precise calibration of the experimental attenuation of drive power.An accurate comparison of numerical results to experiment data as a function of photon number is therefore challenging.For this reason, we do not show the numerical results together with the experimental data in Fig. 5  reach a good trade off between the error of finite separation between pointer states and the error p out of ionizing the transmon.From these histograms P g,e (α, t int ), we compute the overlap (their normalized 2D scalar product) O(t int ) = P g (α, t int )P e (α, t int )dα To compute the average fidelity F = 1 − [P (g|e) + P (e|g)]/2 with P (x|y) the probability to measure x when state y was prepared, we project the dataset onto the imaginary axis.Defining the threshold q th = 0, we compute the error probabilities P (g|e) = P (Im(α) > q th |e) = (1.07 ± 0.14) % and P (e|g) = P (Im(α) ≤ q th |g) = (0.23 ± 0.14) %.We thus obtain F = (99.35± 0.14) %.The uncertainty ±0.14 % comes from the finite number N = 10 6 of repetitions.The error of 0.65 % in the average infidelity is mostly explained by the following processes.By fitting the distribution with double Gaussians [7], we compute the error due to finite separation (∼ 0.13%) between the two Gaussians corresponding to the two states of the qubit.In these experiments, the qubit is first prepared in the ground state |g⟩ using measurement-based feedback with the usual dispersive readout.Using a second standard dispersive readout, we estimate the error of wrong preparation in the ground state before the arming step as P disp (e|g) ∼ 0.2%.Incorrect excited state preparation is also explained by imperfect π pulse due to the coherence limit of the qubit (giving in average, an error of ∼ 0.08%, see Fig. 5 of the main text).Error due to relaxation during the measurement correspond to ∼ 1 2 (1 − e −tint/2T1 ) ∼ 0.2%.

I. Pre-arming amplitude and phase optimization
Because the cavity does not respond at the same frequency during the arming and release steps, the amplitude and phase of the arming step needs to be optimized to obtain the fastest separation between cavity states during readout.Indeed, during the arming step, the steady-state amplitude is given by α s a = (ε 1,a e iϕ1,a /2)/(ω r − ω 1 − iκ/2) while in the release step, it is given by α s r,i = (ε 1,r e iϕ1,r /2)/(ω r + χ i − ω 1 − iκ/2) and depends on the state i of the qubit.For our parameters and choosing ω 1 = ωr + (χ g + χ e )/2, driving at the same amplitude and phase leads to an amplitude ratio of |α s a |/|α s r | = 0.58 and phase difference ϕ α s a − ϕ α s r = 0.154 × 2π where ϕ α s r = (ϕ α s r,g + ϕ α s r,e )/2 is the average phase obtained for the two states of the qubit.
To optimize the readout for the arm-and-release protocol, we measure the overlap error ratio O A&R /O disp between the arm-and-release O A&R and the standard dispersive O disp readouts (Supplementary Figure 5).The chosen arming amplitude and phase is the one minimizing the error ratio O A&R /O disp .

J. Randomized benchmarking under cavity drives
The gate errors under drives in Fig. 5 of the main text are estimated using randomized benchmarking [8].For that purpose, we compare the fidelity of different pulse sequences (Supplementary Figure 6a).In a reference pulse sequence, a number N C of random Clifford gates is applied, followed by a recovery gate before reading out the qubit state.The sequence fidelity F S is fitted using The same fitting procedure is applied for an interleaved sequence where the gate under test is interleaved with the random gates, resulting in the probability p G (see Supplementary Figure 6b).The average gate error is then extracted as ϵ X = 1 2 (1 − p G p G,ref ).

K. Sample and measurement setup
The large features of the sample are made by optical lithography on a Tantalum thin film on a Sapphire substrate, while the Josephson junction of the transmon qubit is fabricated via electronic lithography followed by angle deposition of Al/AlOx/Al in a Plassys evaporator.The readout mode is a λ/4 coplanar waveguide resonator.The Purcell filter is also a λ/4 coplanar waveguide resonator, inductively coupled to the readout mode, and is used as a bandpass filter around the readout frequency.
The sample is cooled down to 10 mK in a dilution refrigerator.The diagram of the microwave wiring is given in Supplementary Figure 7.The qubit, readout and cancellation pulses are generated by modulation of continuous microwave tones produced respectively by generators Anapico APSIN20G and Anapico APSIN12G (readout and cancellation tones use the same local oscillator).They are modulated via IQ-mixers where the intermediate frequency (a few tens of MHz) modulation pulses are generated by 6 channels of an OPX from Quantum Machines with a sample rate of 1 GS/s.The acquisition is performed, after down-conversion by its local oscillator, by digitizing the 100 MHz signal with the 1 GS/s ADC of the OPX.The qubit and cancellation pulses are multiplexed into a single transmission line using a diplexer at the lowest temperature stage.

SUPPLEMENTARY NOTE 3 -POTENTIAL LIMITATIONS L. The need of two driving ports
As previously indicated, qubit cloaking necessitates distinct driving ports-one to address the nonlinear mode and another to address the cavity.In the case of circuit QED devices, this requirement is typically fulfilled with the presence of a port for logical operation on the qubit and a second port for readout on the measurement cavity.In other cavity-based platforms, a single driving port may be easily accessible.With distinct driving ports, a possible limitation is that the optimization-in terms of attenuation of thermal noise, dynamical range, or filtering-of the input transmission line for its use in driving the qubit might compete with its optimization for its use for cloaking.This is particularly true because of the requirement to drive the qubit port at the cavity frequency, and relatively large drive amplitudes necessary b Supplementary Figure 6.X gate randomized benchmarking.a, Pulse sequences used for the randomized benchmarking leading to Fig. 5 of the main text: Top, reference sequence; middle, interleaved sequence; bottom, interleaved sequence under arming drives.b, Dots: measured sequence fidelity (probability to end in the same state as the initial one) as a function of circuit depth NC .Lines: fits using FS = Ap N C G + B. Blue: reference sequence, orange: interleaved without drives, red: interleaved with E1 drive only, and green: interleaved with E1 and E2 drives.The drive ε1/2π = 6.4 MHz corresponds to 0.37 photons on average without cancellation and 0.14 photons with cancellation.
for cloaking.

M. Qubit cloaking in the ultrastrong coupling regime
In the ultrastrong coupling regime, where the qubitcavity coupling is comparable to the cavity and qubit transition frequencies, the dissipator no longer assumes the form in Supplementary Equation (1) but rather takes a correlated qubit-cavity form [9,10].The assumption of separate dissipation channels for the qubit and cavity is no longer a good approximation and the cloaking drive on the qubit cannot cancel the incoherent effects induced by this correlated dissipation.N. The qubit cannot be protected from the thermal excitations in the cavity While the qubit can be cloaked from the coherent drive on the cavity even in the presence of thermal excitations in the cavity, it cannot be cloaked from the cavity thermal excitations themselves: a coherent tone cannot cloak an incoherent excitation.
FIG.2.Vacuum Rabi oscillations in a filled cavity.a, Red and black solid lines correspond to damped vacuum Rabi oscillations of the resonant cavity and qubit, respectively, in the absence of any drive.The qubit-cavity Jaynes-Cummings coupling rate g ′ is set to 100/7 of the cavity decay rate κ.With a cavity drive ϵ1 = 46κ/7 and the cancellation turned on, the cavity field (dashed orange) oscillates on top of |αt| 2 (dasheddotted gray), swapping a single quantum of excitation back and forth with the qubit (dashed light blue).b, Computed Wigner distributions of the cavity field at 3/2 and 1 Rabi periods with the drives on, as indicated by the symbols in a.At 1 Rabi period, the cavity is in a coherent state, while it is in a displaced Fock state at 3/2 Rabi periods.
FIG.4.Arm and release qubit readout.a, Full-cosine numerical simulation of the phase-space trajectories of the cavity field in standard dispersive readout, i.e., square pulse, (dashed line) and the arm and release readout approach (full lines) with an arm-step coherent state α = 2.8i.The path of the arming step is here chosen to reproduce the experiment of panel b, but it can be tailored at will.The coloured dots indicate different times in the evolution.In both readout approaches, the qubit is prepared in the ground state and a π pulse is applied (red lines) or not (blue lines) before the readout step.Simulation parameters are the same as in Fig.3b.b, Experimental qubit measurement error obtained from the overlap between the distributions of the accumulated heterodyne signal over 10 6 repetitions of the experiment.Full line: arm-and-release approach.Dashed line: standard dispersive measurement.c, Arm-and-release pulse sequence used to obtain panels a and b.The arm phase (t < 0) is absent in the case of dispersive readout.

C
. Lledó and A. Moulinas performed the calculations.R. Dassonneville performed the measurements.R. Shillito provided codes.C. Lledó, R. Dassonneville, A. Moulinas, J. Cohen, A. Bienfait, B. Huard, and A. Blais analyzed and discussed the results.C. Lledó, R. Dassonneville, and A. Blais wrote the manuscript with comments from all the authors.A. Bienfait, B. Huard, and A. Blais supervised the work.COMPETING INTERESTS B. H. is an equity shareholder of the Alice&Bob company.The remaining authors declare no competing interests.

Supplementary Figure 4 .
Measured Husimi distributions for standard dispersive and arm-and-release readout.Histograms of the demodulated heterodyne signal for tint = 196 ns and ε1/2π = 63.7 MHz for the standard dispersive readout (a and b) and for the arm-and-release readout (c and d) when the qubit is prepared in its ground state (a and c) or in its excited state (b and d).

Supplementary Figure 5 .
Readout overlap error comparison for standard disperive and arm-and-release methods.Error ratio O A&R /O disp between arm-and-release O A&R and standard dispersive O disp readout errors as function of arming amplitude |α s a | and phase ϕαs a .The overlap errors are extracted for a 140 ns integration time.

Supplementary Figure 7 .
Schematic of the microwave setup.
2. Vacuum Rabi oscillations in a filled cavity.a, Red and black solid lines correspond to damped vacuum Rabi oscillations of the resonant cavity and qubit, respectively, in the absence of any drive.The qubit-cavity Jaynes-Cummings coupling rate g ′ is set to 100/7 of the cavity decay rate κ.With a cavity drive ϵ1 = 46κ/7 and the cancellation turned on, the cavity field (dashed orange) oscillates on top of |αt| 2(dasheddotted gray), swapping a single quantum of excitation back and forth with the qubit (dashed light blue).b, Computed Wigner distributions of the cavity field at 3/2 and 1 Rabi periods with the drives on, as indicated by the symbols in a.At 1 Rabi period, the cavity is in a coherent state, while it is in a displaced Fock state at 3/2 Rabi periods.